A372309
The smallest number whose prime factor concatenation, when written in base n, contains all digits 0,1,...,(n-1).
Original entry on oeis.org
2, 6, 38, 174, 2866, 11670, 135570, 1335534, 15618090, 155077890, 5148702870, 31771759110, 774841780230, 11924858870610, 253941409789410, 3867805835651310
Offset: 2
The factorizations to a(12) are:
a(2) = 2 = 10_2, which contains all digits 0..1.
a(3) = 6 = 2 * 3 = 2_3 * 10_3, which contain all digits 0..2.
a(4) = 38 = 2 * 19 = 2_4 * 103_4, which contain all digits 0..3.
a(5) = 174 = 2 * 3 * 29 = 2_5 * 3_5 * 104_5, which contain all digits 0..4.
a(6) = 2866 = 2 * 1433 = 2_6 * 10345_6, which contain all digits 0..5.
a(7) = 11670 = 2 * 3 * 5 * 389 = 2_7 * 3_7 * 5_7 * 1064_7, which contain all digits 0..6.
a(8) = 135570 = 2 * 3 * 5 * 4519 = 2_8 * 3_8 * 5_8 * 10647_8, which contain all digits 0..7.
a(9) = 1335534 = 2 * 3 * 41 * 61 * 89 = 2_9 * 3_9 * 45_9 * 67_9 * 108_9, which contain all digits 0..8.
a(10) = 15618090 = 2 * 3 * 5 * 487 * 1069, which contain all digits 0..9. See A058909.
a(11) = 155077890 = 2 * 3 * 5 * 11 * 571 * 823 = 2_11 * 3_11 * 5_11 * 10_11 * 47a_11 * 689_11, which contain all digits 0..a.
a(12) = 5148702870 = 2 * 3 * 5 * 151 * 1136579 = 2_12 * 3_12 * 5_12 * 107_12 * 4698ab_12, which contain all digits 0..b.
-
from math import factorial
from itertools import count
from sympy import factorint
from sympy.ntheory import digits
def a(n):
for k in count(factorial(n)):
s = set()
for p in factorint(k): s.update(digits(p, n)[1:])
if len(s) == n: return k
print([a(n) for n in range(2, 10)]) # Michael S. Branicky, Apr 28 2024
A138165
Prime numbers that contain each of the digits 0,1,4,6,8,9 exactly once.
Original entry on oeis.org
104869, 108649, 140689, 140869, 148609, 164089, 164809, 168409, 184609, 186049, 401689, 406981, 408169, 408691, 409861, 416089, 418069, 460189, 460891, 460981, 468019, 468109, 469801, 480169, 486091, 489061, 498061, 601849, 604189, 604819
Offset: 1
-
Select[Prime[Range[10000,50000]],SequenceCount[DigitCount[#],{1,,,1,,1,,1,1,1}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 07 2020 *)
A213737
Odd numbers whose set of prime factors (taken with multiplicity) uses each digit from 0 to 9 exactly once.
Original entry on oeis.org
42279945, 42315045, 42514845, 43092645, 43767645, 45981645, 46149045, 46321845, 52226745, 52654695, 53159595, 56789745, 56841045, 57321645, 58193745, 59869345, 61277145, 61421595, 61860445, 62146545, 62866645, 62936295, 62969845, 63395295, 63411595
Offset: 1
42279945 = 3*5*1049*2687 is in the sequence since the set {3, 5, 1049, 2687} can be formed from the digits 0 to 9 and each digit is used only once.
-
lst = {}; Do[If[Equal[Sort@Flatten@IntegerDigits@FactorInteger[n][[All, {1}]], {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}] && SquareFreeQ[n], AppendTo[lst, n]], {n, 4*10^7 + 1, 7*10^7, 2}]; lst
ed1Q[n_]:=Module[{fi=FactorInteger[n]},Max[Transpose[fi][[2]]]==1 && Union[ Flatten[IntegerDigits/@Transpose[fi][[1]]]]==Range[0,9]]; Select[Range[ 4*10^7+ 1,6.4*10^7,2 ],ed1Q] (* Harvey P. Dale, Dec 19 2014 *)
A290385
Base-ten pandigital factorization integers. The normal factorization (primes raised to greater-than-one exponents) of these numbers uses each digit exactly once.
Original entry on oeis.org
15618090, 20824120, 22022490, 22816290, 22908090, 23294190, 23427135, 23507490, 24843120, 26104560, 26152080, 26679990, 27114690, 27687090, 28275690, 29218704, 29363320, 29447898, 29544690, 29582490, 29670378, 29688144, 29910138, 30120144
Offset: 1
20824120 is in the sequence because 2^3*5*487*1069 is pandigital.
-
pop[d_, mn_] := Union @@ Table[ Select[ FromDigits /@ Flatten[ Permutations /@ Subsets[d, {k}], 1], # > mn && PrimeQ[#] &], {k, IntegerLength@ mn, Length[d]}]; ric[w_, d_, p_] := If[d == {}, cnt++; If[Max[Last /@ w] < 30 && Times @@ (Power @@@ w) <= 4*10^7, AppendTo[L, w]], Block[{pp = pop[d, p], v}, Do[v = Complement[d, IntegerDigits@ x]; ric[Append[w, {x, 1}], v, x]; Do[If[e > 1, ric[Append[w, {x, e}], Complement[v, IntegerDigits@e], x]], {e, Union[ FromDigits /@ Flatten[ Permutations /@ Subsets[v, {1, Length@v}], 1]]}], {x, pp}]]]; Monitor[cnt = 0; L = {}; ric[{}, Range[0, 9], 1];, cnt]; Print["cnt = ", cnt]; Sort[(Times @@ (Power @@@ #)) & /@ L] (* Giovanni Resta, Jul 29 2017 *)
A370612
The smallest number whose prime factor concatenation, when written in base n, does not contain 0 and contains all digits 1,...,(n-1) at least once.
Original entry on oeis.org
3, 5, 14, 133, 706, 2490, 24258, 217230, 2992890, 24674730, 647850030, 4208072190, 82417704810
Offset: 2
a(2) = 3 = 3 whose prime factor in base 2 is: 11.
a(3) = 5 = 5 whose prime factor in base 3 is: 12.
a(4) = 14 = 2*7 whose prime factors in base 4 are: 2, 13.
a(5) = 133 = 7*19 whose prime factors in base 5 are: 12, 34.
a(6) = 706 = 2*353 whose prime factors in base 6 are: 2, 1345.
a(7) = 2490 = 2*3*5*83 whose prime factors in base 7 are: 2, 3, 5, 146.
a(8) = 24258 = 2*3*13*311 whose prime factors in base 8 are: 2, 3, 15, 467.
a(9) = 217230 = 2*3*5*13*557 whose prime factors in base 9 are: 2, 3, 5, 14, 678.
a(10) = 2992890 = 2*3*5*67*1489.
a(11) = 24674730 = 2*3*5*19*73*593 whose prime factors in base 11 are: 2, 3, 5, 18, 67, 49a.
a(12) = 647850030 = 2*3*5*19*1136579 whose prime factors in base 12 are: 2, 3, 5, 17, 4698ab.
a(13) = 4208072190 = 2*3*5*7*61*89*3691 whose prime factors in base 13 are: 2, 3, 5, 7, 49, 6b, 18ac.
a(14) = 82417704810 = 2*3*5*7*23*937*18211 whose prime factors in base 14 are: 2, 3, 5, 7, 19, 4ad, 68cb.
-
from math import factorial
from itertools import count
from sympy import primefactors
from sympy.ntheory import digits
def A370612(n): return next(k for k in count(max(factorial(n-1),2)) if 0 not in (s:=set.union(*(set(digits(p,n)[1:]) for p in primefactors(k)))) and len(s) == n-1)
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